Basically in the EU, you usually have an ID card (or a passport/driving license/visa card, they're recorded on all of those too) that has a combination of a citizen ID and a document ID. Both of these details are combined considered to be "you" for the purposes of anything to do with the government. The government has a registration of every citizen ID+document ID combination and knows as a result what documents are in circulation. They're technically not required in most of Europe, although you must be able to procure one at request for legal reasons (ie. getting your employment properly sorted, opening a bank account, or a law enforcement official asking for your identity). Revoking a combination is as easy as getting a new ID card/passport since the combination is what counts. ID documents also usually expire eventually, so there's also an inherent time limit to what a leaked combination can cause issues with.

They're also as I understand it, used to handle things like sending everyone voter IDs for elections in advance; this is how the government knows who to send the voting cards to.

Bafflingly, the US does NOT have a national identification method that works like this. There's no country-wide identity document that provides the same assurances. As a result, most US entities (government branches & corporations) have settled on a "closest possible"... which is the social security number. A number that's used to identify every person with attachment to the US in some form since social security is something every US citizen has to interact with. (It also includes a ton of non-citizens since as I understand it, social security is something foreign workers also have to interact with, but that's besides the point.) It's a 9 character long numeric string that identifies you as a person... and has almost no revocation mechanism, even if it ends up in a data breach.

Yet in spite of this, it's still used as a country-wide ID mechanism for a lot of different things and replacing it with a proper ID mechanism has as I understand it (not American) very poor support as it's a culture war issue.